Spectral Properties of Positive Maps on C^* -Algebras

Abstract

Perron-Frobenius type results are proved for discrete, Markovian, quantum stochastic processes. IntroductionAt the beginning of the century, Perron [19] and Frobenius [9,10] discovered many important spectral properties possessed by matrices with positive entries. There now exists a vast literature extending some of their results to positive operators on a large class of ordered vector spaces, the most successful results being with compact operators and/or cones with a lattice ordering or a large interior. We refer the reader to [11,15,22,32] and the references quoted therein. Here we regard the original Perron-Frobenius theory as being concerned with the spectral properties of positive operators on finite dimensional, commutative C*-algebras, and give a noncommutative version of this theory, at least for finite-dimensional C*-algebras.The first part of the Perron-Frobenius story tells us that the spectral radius of a positive matrix with positive entries is an eigenvalue, possessing a positive eigenvector. Moreover, if the matrix is irreducible in a certain sense, then the spectral radius is a simple eigenvalue, and (apart from scalar multiplication) there are no other positive eigenvectors. It is this part of the theory which has received most attention by other authors, referred to above. In §2 we give our generalization of this to finite-dimensional C*-algebras.Perron and Frobenius also showed that the spectrum and eigenvectors of an irreducible positive matrix had certain multiplicative properties. This part of the theory has not received nearly as much attention, although Rota [20] and Schaefer [21] obtained some results in this direction for certain lattice ordered spaces, namely IP-spaces and commutative C*-algebras. Analogous results were obtained by St0rmer for ergodic groups of automorphisms on von Neumann algebras [29]. In §3 and §4 we study multiplicative properties associated with the spectrum of an irreducible positive operator on a finite-dimensional C*-algebra. In §3, it is mainly the Jordan structure which is important, but the C*-structure takes over in §4 when we restrict attention to those maps which satisfy the Schwarz inequality which we call Schwarz maps.In [4] and [5] Davies has proposed concepts of recurrence and transience for certain continuous-time Markovian quantum stochastic processes. In §3 we also propose definitions of recurrence and transience for discrete-time Markovian quantum stochastic processes, which are different from those of Davies. (Davies's ideas easily carry over from continuous time to discrete time.) Discrete non-Markovian quantum stochastic processes have been studied recently by Accardi [1,2] and Lindblad [16].

Random repeated quantum interactions and random invariant states

“…Irreducible unital maps which satisfy the Schwarz inequality have very nice peripheral spectra. The proof of the following important result can be found in one of [11,13,15], in more general settings. Irreducible (and, in particular, strictly positive) quantum channels have desirable spectral properties, hence the interest one has for these classes of maps.…”

“…In fact, the following characterization of irreducibility is known [11]. Irreducible unital maps which satisfy the Schwarz inequality have very nice peripheral spectra.…”

Irreducible positive linear maps on operator algebras

“…Therefore, the classical matrix theoretic results (see [7,Ch.8]) of O. Perron and G. Frobenius on matrices with nonnegative entries can be viewed as an important case of a more general theory that deals with positive maps on operator algebras. This is the viewpoint taken by D. Evans and R. Høegh-Krohn [6], S. Albeverio and R. Høegh-Krohn [2], and U. Groh [8], [9] in their works on the spectra of positive maps on operator algebras. In both the classical theory and its various generalisations [12], and in the one put forward in [6], [2], [8], and [9], a positive map that is "strictly" positive, or that is "irreducible", will possess certain interesting spectral properties.…”

“…This is the viewpoint taken by D. Evans and R. Høegh-Krohn [6], S. Albeverio and R. Høegh-Krohn [2], and U. Groh [8], [9] in their works on the spectra of positive maps on operator algebras. In both the classical theory and its various generalisations [12], and in the one put forward in [6], [2], [8], and [9], a positive map that is "strictly" positive, or that is "irreducible", will possess certain interesting spectral properties. Although the spectral theory of such maps has been studied, the issue of how one is to determine whether a given positive linear map on an operator algebra is strictly positive or irreducible (or neither) has received much less attention.…”

“…First of all we aim to develop a "usable" criterion for the irreducibility of certain positive maps on operator algebras; this is done in order to facilitate the construction of concrete irreducible positive maps to accompany the more theoretical works [2], [6], [8] on spectral theory. The second point to be made here is that the notion of irreducibility for positive maps on operator algebras is combinatorial, just as it is for nonnegative matrices; this fact is of interest because, in general, cone-preserving maps are not usually explicitly described by combinatorial data.…”